An Adaptive Finite Element Method for the Diffraction Grating Problem with PML and Few-Mode DtN Truncations

Zhou, Weiqi; Wu, Haijun

doi:10.1007/s10915-018-0683-0

Cited by 18 publications

(16 citation statements)

References 39 publications

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“…Unlike the exponential convergence results in [10,34], (43) indicates only a poor convergence of the PML method over S H . We however believe that exponential convergence can be realized in a compact subset of S H , which is indeed true if Γ is flat [6].…”

Section: Semi-waveguide Problemsmentioning

confidence: 84%

“…If the surface has no defects, the total wave field for the plane-wave incidence is quasiperiodic so that the original scattering problem can be formulated in a single unit cell, bounded laterally but unbounded vertically. According to UPRC, the scattered wave at infinity can then be expressed in terms of upgoing Bloch waves, so that a transparent boundary condition or PML of a local/nonlocal boundary condition can be successfully used to terminate the unit cell vertically; readers are referred to [3,10,26,34] and the references therein, for related numerical methods as well as theories of exponential convergence due to a PML truncation. But, if the incident wave is nonquasi-periodic, e.g., the cylindrical wave, or if the surface is locally defected, much fewer numerical methods or theories have been developed as it is no longer straightforward to laterally terminate the scattering domain.…”

Section: Introductionmentioning

confidence: 99%

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PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface

Yu¹,

Hu²,

Lu³

et al. 2021

Preprint

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This paper studies the perfectly-matched-layer (PML) method for wave scattering in a half space of homogeneous medium bounded by a two-dimensional, perfectly conducting, and locally defected periodic surface, and develops a high-accuracy boundary-integral-equation (BIE) solver. Along the vertical direction, we place a PML to truncate the unbounded domain onto a strip and prove that the PML solution converges linearly to the true solution in the physical subregion of the strip with the PML thickness. Laterally, we divide the unbounded strip into three regions: a region containing the defect and two semi-waveguide regions, separated by two vertical line segments. In both semi-waveguides, we prove the well-posedness of an associated scattering problem so as to well define a Neumann-to-Dirichlet (NtD) operator on the associated vertical segment. The two NtD operators, serving as exact lateral boundary conditions, reformulate the unbounded strip problem as a boundary value problem onto the defected region. Due to the periodicity of the semi-waveguides, both NtD operators turn out to be closely related to a Neumann-marching operator, governed by a nonlinear Riccati equation. It is proved that the Neumann-marching operators are contracting, so that the PML solution decays exponentially fast along both lateral directions. The consequences culminate in two opposite aspects. Negatively, the PML solution cannot exponentially converge to the true solution in the whole physical region of the strip. Positively, from a numerical perspective, the Riccati equations can now be efficiently solved by a recursive doubling procedure and a high-accuracy PML-based BIE method so that the boundary value problem on the defected region can be solved efficiently and accurately. Numerical experiments demonstrate that the PML solution converges exponentially fast to the true solution in any compact subdomain of the strip.

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Section: Semi-waveguide Problemsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface

Yu¹,

Hu²,

Lu³

et al. 2021

Preprint

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“…Remark 2.2. The recent work [58] introduced the few-mode DtN technique to deal with the evanescent wave components, while the other modes were treated with the classical PML n .…”

Section: Andmentioning

confidence: 99%

A Truly Exact Perfect Absorbing Layer for Time-Harmonic Acoustic Wave Scattering Problems

Yang¹,

Wang²,

Gao³

2021

SIAM J. Sci. Comput.

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In this paper, we design a truly exact perfect absorbing layer (PAL) for domain truncation of the two-dimensional Helmholtz equation in an unbounded domain with bounded scatterers. This technique is based on a complex compression coordinate transformation in polar coordinates and a judicious substitution of the unknown field in the artificial layer. Compared with the widely used perfectly matched layer (PML) methods, the distinctive features of PAL lie in that (i) it is truly exact in the sense that the PAL-solution is identical to the original solution in the bounded domain reduced by the truncation layer; (ii) with the substitution, the PAL-equation is free of singular coefficients and the substituted unknown field is essentially nonoscillatory in the layer; and (iii) the construction is valid for general star-shaped domain truncation. By formulating the variational formulation in Cartesian coordinates, the implementation of this technique using standard spectralelement or finite-element methods can be made easy as a usual coding practice. We provide ample numerical examples to demonstrate that this method is highly accurate and robust for very high wavenumbers and thin layers. It outperforms the classical PML and the recently advocated PML using unbounded absorbing functions. Moreover, it can fix some flaws of the PML approach.

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“…However, Singer and Turkel [84] demonstrated that such a transformation can magnify the evanescent waves in the waveguide setting. Recently, Zhu and Wu [98] proposed a mixed PML and (few-mode) DtN truncation (to deal with the evanescent modes) for the waveguide related to diffraction grating problems. Moreover, as reported in [60,65,36], the PML has flaws and failures at times.…”

Section: Existing Techniques For Domain Reductionmentioning

confidence: 99%

“…Remark 2.3. To overcome the failure of PML for the evanescent waves, Zhu and Wu [98] proposed a mixed PML and (few-mode) DtN truncation (to deal with the evanescent modes) for the waveguide related to diffraction grating problems.…”

Section: The Pml Technique and Its Error Analysismentioning

confidence: 99%

Perfect absorbing layers for wave scattering problems and related topics

Gao¹

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List of Figures 2.3 Profiles of the PML solution U p with regular function (2.14) (σ 0 = 1, n = 2) and singular function (2.15) vs PAL solution U a in Ω and V a in Ω d with k = 99.99, (L, d) = (1, 0.2) along the x-axis at y = π/2. Left: real part solution; Right: imaginary part solution.. .. .. .. .. . 2.4 Left: illustration of the circular PAL from radial direction. Middle: schematic illustration of the rectangular PAL Ω d with four trapezoidal patches. Right: illustration of the elliptic PAL under elliptic coordinate. 2.5 Left: errors against various N for different layers (width d = 0.1) with boundary condition g(y) = sin(5y) − sin(30y); Right: errors against various N for different layers (width d = 0.5) with boundary condition g(y) = sin(5y) − sin(30y).

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An Adaptive Finite Element Method for the Diffraction Grating Problem with PML and Few-Mode DtN Truncations

Cited by 18 publications

References 39 publications

PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface

PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface

A Truly Exact Perfect Absorbing Layer for Time-Harmonic Acoustic Wave Scattering Problems

Perfect absorbing layers for wave scattering problems and related topics

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